Convergence acceleration of some continued fractions (Q1397928)

This rather lengthy paper contains a plethora of formulae connected with convergence acceleration of continued fractions \[ \text{K}_{n=1}^{\infty} {a_n\over b_n} \] with tails \[ t_n=\text{K}_{j=n+1}^{\infty} {a_j\over b_j} (n\geq 0), \] satisfying the conditions C1 or C2: C1: \(a_n,b_n\) are rational functions of \(n\) for \(n\) sufficiently large and \[ t_n\sim \sum_{i=0}^{\infty} q_i (n+1)^{\mu -i},\quad q_0\not= 0 \] with \(\mu\in\text{Z}\) and \(\sim\) indicating that the series need not converge, C2: the condition on \(a_n,b_n\) and the tails \(t_n\) again hold, but with the result depending on the parity of \(n\) (i.e. for \(t_n\) the \(\mu,q_i\) may be different for \(n\) odd or even). The author coins the phrase one-variant resp. two-variant for C1 resp. C2 (actually limit-periodicity). In both cases the tails can be seen to satisfy a bilinear equation \[ e_nt_{n-1}t_n+f_nt_{n-1}+g_nt_n=h_n \] for certain quantities. The transformed tails then satisfy a similar equation where all quantities on the left hand side get a suffix \({}^{(k)}\) and the right hand side is \(e_n^{(k-1)}\); the transformation is described below. Let \(u_n^{(k)}\) be the effectively principal part of the tail \(t_n^{(k)}\), i.e. a polynomial in \(n\) with \(t_n^{(k)} - u_n^{(k)}=o(1)\), leading to the transformation rule \[ t_n^{(k)}=u_n^{(k)}+1/t_n^{(k+1)},\quad k\geq 0. \] Several theorems concerning connections between the coefficients \(e_n^{(k)}, f_n^{(k)}, g_n^{(k)}\) are then given for the cases where these quantities are polynomials in \(n\) of degrees \(\xi, \lambda, \nu\) resp. for the triples \((0,0,1)\), \((0,1,1,)\), \((0,2,2)\), \((1,2,2)\), \((1,3,3)\) and \((0,3,3)\). Several known results are recovered and in only a few cases the acceleration of the convergence is proved explicitly, while in other cases numerical evidence is given. In total the paper makes a very crowded impression and the choice of the examples is not always given a specific reason but, nevertheless, interesting reading.